MAC 1105
Final Review
Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational
expression.
8x 2 - 49x + 6
1)
x-6
A)
1
,x
x-6
6
B) 8x - 1, x
D)
8x 2 - 49x + 6
,x
x-6
(x + 3)2
(x - 3)2
C)
6(x2 + 9)
x2 - 9
D)
16
x
C)
x
16
D) 16
C) 8x 2 - 50, no restrictions on x
Divide.
2)
6
6
(x + 3)2 x 2 - 9
÷
x-3
3x - 9
A)
(x + 3)3
3(x - 3)
B)
3(x + 3)
x-3
Simplify the complex rational expression.
2
4+
x
3)
x 1
+
4 8
A) 1
B)
Solve the equation.
2
8
4
=
4)
y + 5 y - 5 y2 - 25
A) {54}
B) {9}
C) { 37}
Perform the indicated operations and write the result in standard form.
5) -5 - -121
A) i( 5 + 11)
B) 5i - 11
C) 5i - 11i
Find the product and write the result in standard form.
6) (3 + 8i)(3 - 8i)
A) 73
B) 9 - 64i
C) -55
D) {-9}
D) i( 5 - 11)
D) 9 - 64i2
Divide and express the result in standard form.
2
7)
3+i
A)
3 1
- i
4 4
B)
3 1
+ i
4 4
C)
3 1
+ i
5 5
D)
3 1
- i
5 5
Solve the equation by factoring.
8) 25x2 + 30x + 8 = 0
A) -
4
1
,25
4
B)
4
2
,5
5
C)
4 2
,
5 5
D) -
4
2
,5
5
D) ±
2i
5
Solve the equation by the square root property.
9) (x - 5)2 = -4
A) {5i ± 2}
B) {5 ± 2i}
C) {-5 ± 2i}
Solve the equation by completing the square.
10) x2 + 8x - 3 = 0
A) {-4 C) {-1 -
19 , -4 +
19 , -1 +
19}
19}
B) {-4 - 1 19 , -4 + 1 19}
D) {4 + 19}
Solve the equation using the quadratic formula.
11) 4x 2 + x - 4 = 0
A)
C)
1-
8
65 1 +
,
8
65
B)
-1 - 65 -1 + 65
,
2
2
-1 - 65 -1 + 65
,
8
8
D)
Compute the discriminant. Then determine the number and type of solutions for the given equation.
12) x2 + 6x - 7 = 0
A) 0; one real solution
B) -8; two complex imaginary solutions
C) 64; two unequal real solutions
Solve the equation,.
13) x - 3x - 2 = 4
A) {-1}
Solve the equation.
14) x4 - 40x2 + 144 = 0
A) {-2, 2, -6, 6}
Solve the inequality.
15) 5x + 4 < 4
A) - , -
16) 4x - 2
8
A) - , -
B) {1, 2}
C) {9}
D) {2, 9}
B) {2, 6}
C) {-2i, 2i, -6i, 6i}
D) {4, 36}
8
5
0,
B) - , -
8
5
5
2
[8, )
B) - , -
3
2
5
,
2
C) -
8
,0
5
D) (- , 5)
C) -
3 5
,
2 2
5
D) [ , )
2
Determine whether the relation is a function.
17) {(-8, -9), (-8, 9), (1, 3), (3, 5), (10, -9)}
A) Not a function
Determine whether the equation defines y as a function of x.
18) x2 + y = 16
A) y is a function of x
B) Function
B) y is not a function of x
Use the graph to determine the function's domain and range.
19)
A) domain: [0, )
range: (- , )
B) domain: (- , )
range: [-1, )
C) domain: [0, )
range: [-1, )
D) domain: [0, )
range: [0, )
C) (-2, )
D) (-2, 2)
Identify the intervals where the function is
20) Increasing
A) (-3, 3)
B) (-3, )
Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin.
21) y = 3x 2 - 3
A) y-axis only
C) x-axis only
B) Origin only
D) x-axis, y-axis, origin
Determine whether the given function is even, odd, or neither.
22) f(x) = -5x5 + x3
A) Even
B) Odd
Evaluate the piecewise function at the given value.
23) f(x) = 4x + 3 if x < 2 ; f(3)
5x + 2 if x 2
A) 17
B) 14
Find and simplify the difference quotient
24) f(x) = 3x - 7
-14
A) 3 +
h
f(x + h) - f(x)
,h
h
B) 3 +
C) Neither
C) 20
D) 18
0 for the given function.
6(x - 7)
h
C) 3
D) 0
Find the domain of the function.
2x
25) g(x) =
x2 - 9
A) (- , )
C) (9, )
26)
B) (- , -3) (-3, 3)
D) (- , 0) (0, )
(3, )
x
x-4
A) (- , 4)
(4, )
C) (- , )
D) (4, )
Given functions f and g, perform the indicated operations.
g(x) = -4x + 8
27) f(x) = 8 - 8x,
Find f + g.
A) -4x + 16
B) -4x + 8
C) -12x + 16
D) 4x
For the given functions f and g , find the indicated composition.
28) f(x) = -2x + 2, g(x) = 3x + 2
(g f)(x)
A) 6x + 8
B) -6x + 6
C) -6x + 8
D) -6x - 4
C) 163
D) 251
C) (- , 3) or (3, )
D) (- , )
29) f(x) = x 2 - 2x - 5,
(f g)(-5)
A) 955
B) [4, )
g(x) = x2 + 2x - 1
B) 867
Given functions f and g, determine the domain of f + g.
5
g(x) =
30) f(x) = 3x + 3,
x-3
A) (0, )
B) (- , -5) or (-5, )
Find the inverse of the one-to-one function.
6x - 7
31) f(x) =
5
5x + 7
A) f-1 (x) =
6
5x - 7
B) f-1 (x) =
6
C) f-1 (x) =
5
6x - 7
D) f-1 (x) =
5
6x + 7
Begin by graphing the standard quadratic function f(x) = x 2 . Then use transformations of this graph to graph the given
function.
32) h(x) = (x - 5)2 + 7
A)
B)
C)
D)
Find the distance between the pair of points.
33) (2, 6) and (-4, -7)
A) 78
B)
133
C)
Find the midpoint of the line segment whose end points are given.
34) (6, -6) and (-7, -3)
A) (-1, -9)
B) (13, -3)
C) (-
205
1
9
,- )
2
2
Write the standard form of the equation of the circle with the given center and radius.
35) (7, 0); 7
A) x2 + (y - 7)2 = 7
B) x2 + (y + 7)2 = 7
C) (x + 7)2 + y2 = 49
D) -7
D) (
13
3
,- )
2
2
D) (x - 7)2 + y2 = 49
Find the center and the radius of the circle.
36) (x + 2)2 + (y - 3)2 = 49
A) (-2, 3), r = 7
B) (-3, 2), r = 49
C) (2, -3), r = 49
D) (3, -2), r = 7
Complete the square and write the equation in standard form. Then give the center and radius of the circle.
37) x2 + y2 - 18x + 4y + 85 = 9
A) (x - 9)2 + (y + 2)2 = 9
(-9, 2), r = 9
C) (x + 2)2 + (y - 9)2 = 9
(2, -9), r = 9
B) (x + 2)2 + (y - 9)2 = 9
(-2, 9), r = 3
D) (x - 9)2 + (y + 2)2 = 9
(9, -2), r = 3
Graph the circle.
38) (x - 1)2 + (y - 3)2 = 36
A)
B)
C)
D)
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
39) f(x) = (x + 3)2 - 5
A) (3, 5)
B) (-3, -5)
C) (3, -5)
D) (-3, 5)
40) f(x) = x 2 - 2x - 4
A) (-2, 4)
B) (1, -7)
C) (-1, -1)
D) (1, -5)
Find the axis of symmetry of the parabola defined by the given quadratic function.
41) f(x) = (x + 2)2 + 7
A) y = 7
Find the range of the quadratic function.
42) f(x) = (x + 2)2 + 8
A) [-8, )
B) x = 2
C) x = -2
D) y = -7
B) [8, )
C) [-2, )
D) [2, )
Find the x-intercepts (if any) for the graph of the quadratic function.
43) f(x) = x 2 + 12x + 15 Give your answers in exact form.
A) (6 +
21, 0)
B) (-6 ±
21, 0)
C) (-12 ±
15, 0)
D) (6 ±
15, 0)
Divide using long division.
4m 3 + 21m 2 - 42m + 49
44)
m+7
A) m 2 + 7m + 4
45)
B) 4m 2 + 7m + 7
C) 4m 2 - 7m + 7
D) m 2 + 8m + 9
x 4 + 16
x-2
A) x3 - 2x2 + 4x - 8 +
32
x-2
C) x3 + 2x2 + 4x + 8
B) x3 + 2x2 + 4x + 8 +
32
x-2
D) x3 + 2x2 + 4x + 8 +
16
x-2
Solve the inequality by the test-point method. Write the solution in interval notation.
46) x2 - 3x - 18 < 0
A) (-3, 6)
B) (- , -3)
C) (- , -3)
(6, )
D) (6, )
Solve the rational inequality. Write the solution in interval notation.
x
<3
47)
x-4
A) (-4, 4)
B) 4, 6
C) (- , 4)
6,
D)
Use the compound interest formulas to solve.
48) Find the accumulated value of an investment of $6000 at 4% compounded semiannually for 8 years.
A) $8236.71
B) $8211.41
C) $7920.00
D) $7029.96
49) Find the accumulated value of an investment of $2000 at 8% compounded continuously for 4 years.
A) $2640.00
B) $2754.26
C) $2720.98
D) $2854.26
50) Find out how long it takes a $3000 investment to double if it is invested at 7% compounded monthly. Round to
the nearest tenth of a year.
A) 10.1 years
B) 9.9 years
C) 10.3 years
D) 9.7 years
Graph the function.
51) Use the graph of f(x) = 2 x to obtain the graph of g(x) = 2 x - 3.
A)
B)
C)
D)
Solve the equation by expressing each side as a power of the same base and then equating exponents.
1
52) 2(3x + 5) =
16
A) {3}
B)
1
8
Write the equation in its equivalent exponential form.
53) log 4 = x
2
A) 2x = 4
B) x2 = 4
Write the equation in its equivalent logarithmic form.
54) 43 = y
A) log
y
B) log
4=3
4
y=3
C) {8}
D) {-3}
C) 42 = x
D) 4x = 2
C) log
3
y=4
D) log
y
3=4
The graph of a logarithmic function is given. Select the function for the graph from the options.
55)
A) f(x) = log
4
x
B) f(x) = 1 - log
Find the domain of the logarithmic function.
56) f(x) = log (x - 8)
3
A) (8, )
B) (-8, )
4
x
C) f(x) = log
4
(-x)
D) f(x) = - log
B) 3
Evaluate the expression without using a calculator.
58) log 12
12
1
A) 1
B)
12
x
C) (- , 8) or (8, )
D) (- , 0) or (0, )
C) -3
D)
C) 12
D) 0
Evaluate the expression.
1
57) log
4 64
A) 12
4
1
3
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions.
59) log (7x)
7
A) 1
B) 7
C) 1 + log x
D) x
7
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
x
60) log
10
A) log x + 1
B) 10x
C) -10x
D) log x - 1
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions.
13x
61) log
w 2
A) log
2
w
C) log 13 + log x + log 2
w
w
w
w
13x - log
B) log
D) log
w
w
13 + log
11x
w
x - log
w
2
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
62) log 390
26
A) 1.1761
B) 4.0060
C) 1.8312
D) 0.5461
Solve the exponential equation. Express the solution set in terms of natural logarithms.
63) 5 x + 6 = 3
A) {ln 3 - ln 5 - ln 6}
B)
ln 5
+6
ln 3
64) e x + 7 = 5
A) {ln 12}
B) {ln 5 - 7}
C)
ln 5
+ ln 6
ln 3
C) {e35}
D)
ln 3
-6
ln 5
D) {e5 + 7}
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
65) log (x + 3) = 3
5
A) {246}
B) {122}
C) {240}
D) {128}
66) log x + log (x - 3) = 2
2
2
A) {4}
67) log (x + 5) = log (5x - 4)
9
A)
4
B) {2}
B)
1
4
C) {-1, 4}
C) -
9
4
D) {1, -4}
D)
3
2
Solve the system of equations by the substitution method.
68)
y = 4x - 3
2y + 8x = 26
A) {(5, 2)}
B) {(-2, 5)}
C) {(2, 5)}
D)
Solve the system by the elimination by addition method.
69) 7x + 8y = -19
4x - 3y = -26
A) {(-5, 3)}
B) {(-5, 2)}
C) {(-6, 3)}
D)
Solve the system by the method of your choice.
70) y = 18 - 6x
6x + y = 54
A) {(12, 6)}
C) {(x, y) 6x + y = 18}
B) {(10, 12)}
D)
Solve the system of equations.
71) x + y + z = 2
x - y + 2z = -3
2x + y + z = 0
A) {(1, -2, 3)}
C) {(3, -2, 1)}
B) {(1, 3, -2)}
D) {(-2, 3, 1)}
Solve the problem.
72) A vendor sells hot dogs, bags of potato chips, and soft drinks. A customer buys 5 hot dogs, 4 bags of potato
chips, and 5 soft drinks for $17.00. The price of a hot dog is $1.25 more than the price of a bag of potato chips.
The cost of a soft drink is $2.25 less than the price of two hot dogs. Find the cost of each item.
A) $2.00 for a hot dog; $0.75 for a bag of potato chips; $1.25 for a soft drink
B) $1.75 for a hot dog; $0.50 for a bag of potato chips; $1.25 for a soft drink
C) $0.50 for a hot dog; $1.75 for a bag of potato chips; $1.25 for a soft drink
D) $1.75 for a hot dog; $1.25 for a bag of potato chips; $0.50 for a soft drink
73) The Family Fine Arts Center charges $21 per adult and $15 per senior citizen for its performances. On a recent
weekend evening when 525 people paid admission, the total receipts were $8973. How many who paid were
senior citizens?
A) 342 senior citizens
B) 273 senior citizens
C) 183 senior citizens
D) 252 senior citizens
Graph the inequality.
74) x - y > -4
A)
B)
C)
D)
Graph the solution set of the system of inequalities or indicate that the system has no solution.
75) y < -x + 5
y > 2x - 3
A)
B)
C)
D)